Georgian Mathematical Journal Volume 10 (2003), Number 3, 543–548

ON THE ASYMPTOTICS OF SOLUTIONS OF ELLIPTIC EQUATIONS IN A NEIGHBORHOOD OF A CRACK WITH

NONSMOOTH FRONT

V. A. KONDRAT’EV AND V. A. NIKISHKIN

Abstract. Two terms of asymptotics near crack are obtained for solutions of the Dirichlet boundary value problem for second-order elliptic equations in divergent form. The front of a crack is from C^{1+s} and the coefficients of the equations belong to C^{s} (0.5 < s < 1).

2000 Mathematics Subject Classification: 35J25.

Key words and phrases: Asymptotics, solution of elliptic equation, crack, nonsmooth front, Dirichlet boundary value problem, divergent form.

A second-order elliptic equation in G \ Ω is considered, where G ⊂ ℝ^{n} is a domain with a smooth boundary, Ω is an (n − 1)-dimensional manifold with a boundary from C^{1+s}, 0 < s < 1.

We study solutions of the equation

∑_{i,j=1}^{n} \frac{∂}{∂x_{i}} (a_{ij}(x) \frac{∂u}{∂x_{j}} ) = 0,  x ∈ G \ Ω,  (1)

that belong to the Sobolev space W_{2}^{1}(G) and satisfy the condition

u(x) = 0,  x ∈ Ω.  (2)

Let L be the boundary of Ω, r be the distance from x to L, x_{L} be a point on L nearest to x, φ be the polar angle in the plane normal to L and passing through x_{L}.

Singularities of solutions of elliptic equations near a nonsmooth boundary were studied by many authors, see, e.g., [1]–[4].

In [4], the following representation was obtained for the plane (n − 1)-dimen-sional domain Ω:

u(x) = C(x_{L})r^{1/2}Φ(φ) + u_{1}(x),

where Φ(φ) is a smooth function,

|u_{1}| ≤ C_{0}r^{1/2+ε}, 0 < ε < min{s, 0.5},
|C(x_{L})| + |C_{0}| ≤ const · ∥u∥_{W_{2}^{1}(G)}.

The following result is the main theorem of our paper.

Theorem 1. Let u(x) ∈ W_{2}^{1}(G) be a solution of problem (1), (2), and let

a_{ij} ∈ C^{s}(G)  (i, j = 1, …, n),  Ω ∈ C^{1+s},

where 0 < s < 1.

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Then:

if 0 < s ≤ 0.5, then

u(x) = C(x_{L})r^{1/2}Φ(φ) + u_{1}(x),

where Φ(φ) is a smooth function,

|u_{1}| ≤ C_{0}r^{1/2+ε}, 0 < ε < s,
|C(x_{L})| + |C_{0}| ≤ const · ∥u∥_{W_{2}^{1}(G)};

if 0.5 < s < 1, then

u(x) = C_{1}(x_{L})r^{1/2}Φ_{1}(φ) + C_{2}(x_{L})rΦ_{2}(φ) + u_{1}(x),

where Φ_{1}(φ), Φ_{1}(φ) are smooth functions

|u_{1}| ≤ C_{0}r^{1+ε}, 0 < ε < s − 0.5,
|C_{1}(x_{L})| + |C_{2}(x_{L})| + |C_{0}| ≤ const · ∥u∥_{W_{2}^{1}(G)}.

Straightening of the boundary. Let P be an arbitrary point of the set L. Consider a neighborhood U of P in which Ω admits a one-to-one projection to the tangent plane. Assume that P is the origin and that in this neighborhood we have

Ω = {x | x_{1} = F (x_{2}, …, x_{n}), (x_{2}, …, x_{n}) ∈ Ω_{1}}, O ∈ ∂Ω_{1}.

∂Ω_{1} is given in a neighborhood of the origin by the equation

x_{2} = h(x_{3}, …, x_{n}) ∈ C^{1+s}.

Let us extend F (x_{2}, …, x_{n}) ∈ C^{1+s} in a neighborhood of the origin so that the class of smoothness be prescribed. Let F (0), ∇F (0) = 0.

Introduce an averaging kernel K(τ) such that K(τ) ∈ C^{∞}(R^{1}), K(τ) is even, K(τ) ≡ 0 for | τ |≥ 1, and

∫_{−1}^{1} K(τ)dτ = 1.

The straightening of the boundary consists of two steps.

The first transformation of the coordinates has the form:

x_{1} = x′_{1} + H(x′), x_{2} = x′_{2}, …, x_{n} = x′_{n},

where

H(x′) = ∫_{R^{n−1}} F (t) ∏_{l=2}^{n} (\frac{1}{|x_{1}|} K (\frac{t_{l} − z′_{l}}{|x_{1}|} )) dt.

The second transformation of the coordinates is the same as that in [4].

Under the above transformations, equation (1) becomes an equation in divergent form with coefficients in C^{s}.

Dirichlet problem in a dihedral angle for an equation with constant coefficients. Let G_{0} be a dihedral angle

G_{0} = {x | 0 < x_{1}^{2} + x_{2}^{2} < ∞, 0 < φ < ω},

where φ is the polar angle in the plane (x_{1}, x_{2}).

Set

ρ = \sqrt{∑_{i=1}^{n} x_{i}^{2}} , r′ = \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{ρ} .

We need the weighted Sobolev spaces Ẇ_{α,β}^{0} and Ẇ_{α,β}^{1} in which the norms are defined as follows:

∥u∥2 ; Ẇ_{α,β}^{0} = ∫_{G_{0}} u^{2}ρ^{α}(r′)^{β}dx,

∥u∥2 ; Ẇ_{α,β}^{1} = ∫_{G_{0}} ρ^{α}(r′)^{β} grad^{2} udx + ∫_{G_{∘}} u^{2}ρ^{α−2}(r′)^{β−2}dx.

Let u(x) ∈ W_{2}^{1}(G_{0}) be a generalized solution (here and below, in the sense of distributions) of the following Dirichlet problem:

Δu(x) = f_{0}(x) + ∑_{i=1}^{n} \frac{∂f_{i}(x)}{∂x_{i}},  x ∈ G_{0},  (3)
u(x) = 0,  x ∈ ∂G_{0}.  (4)

The following two assertions can be proved by the method developed in [1].

Theorem 2. Let u(x) ∈ W_{2}^{1}(G_{0}) be a generalized solution of problem (3), (4), where

f_{0} ∈ Ẇ_{α,β}^{0}(G_{0}),  f_{i} ∈ Ẇ_{α−2,β−2}^{0}(G_{0})  (i = 1, …, n),

α + 2 (\frac{π}{ω} − 2) + n > 0, β + 2 (\frac{π}{ω} − 2) + n − 1 > 0.

Then u ∈ Ẇ_{α−2,β−2}^{1}(G_{0}).

Theorem 3. Let u(x) ∈ W_{2}^{1}(G_{0}) be a generalized solution of problem (3), (4), where

f_{0} ∈ Ẇ_{α,β}^{0}(G_{0}), f_{i} ∈ Ẇ_{α−2,β−2}^{0}(G_{0}) (i = 1, …, n),

α + 2 (\frac{2π}{ω} − 2) + n < 0, β + 2 (\frac{π}{ω} − 2) + n − 1 > 0,

α + 2 (\frac{3π}{ω} − 2) + n > 0.

Then u(x) can be represented in the form

u(x) = C_{1}ρ^{\frac{π}{ω}} Φ_{1}(θ) + C_{2}ρ^{\frac{2π}{ω}} Φ_{2}(θ) + u_{1},

where θ are the coordinates on the unit sphere, Φ_{1}(θ),Φ_{1}(θ) are the eigenfunctions of the Beltrami operator, and

u_{1} ∈ Ẇ_{α−2,β−2}^{1}(G_{0}).

Dirichlet problem in a dihedral angle for an equation with variable coefficients. Let u(x) ∈ W_{2}^{1}(G_{0}) be a generalized solution of the Dirichlet problem

∑_{i,j=1}^{n} \frac{∂}{∂x_{i}} (a_{ij}(x) \frac{∂u(x)}{∂x_{j}}) = f_{0}(x) + ∑_{i=1}^{n} \frac{∂f_{i}(x)}{∂x_{i}},  x ∈ G_{0},  (5)
u(x) = 0,  x ∈ ∂G_{0},  (6)

where a_{ij} ∈ C^{s}(\overline{G}_{0}), a_{ij}(0) = δ_{ij} (without loss of generality) (i, j = 1, …, n).

Lemma 1. Let u(x) ∈ W_{2}^{1}(G_{0}) be a generalized solution of problem (5), (6), where

f_{0} ∈ Ẇ_{2−2ks+ε_{0},2}^{0}(G_{0}), f_{i} ∈ Ẇ_{−2ks+ε_{0},0}^{0}(G_{0}) (i = 1, …, n),

2 − 2ks + ε_{0} + 2 (\frac{π}{ω} − 2) + n > 0, 2 − 2(k + 1)s + ε_{0} + 2 (\frac{π}{ω} − 2) + n < 0,

k is a nonnegative integer, and ε_{0} > 0 is sufficiently small.

Then

u ∈ Ẇ_{−2ks+ε_{0},0}^{1}(G_{0}).

The proof of Lemma 1 follows from Theorem 2 by induction on k_{1} (0 ≤ k_{1} ≤ k) and is based on the representation of equation (5) in the form

Δu(x) = f_{0}(x) + ∑_{i=1}^{n} \frac{∂}{∂x_{i}} (f_{i}(x) − ∑_{j=1}^{n}(a_{ij}(x) − δ_{ij})\frac{∂u}{∂x_{j}} ) .

Using this representation, Lemma 1, and Theorem 3, we obtain the following assertion.

Lemma 2. Let u(x) ∈ W_{2}^{1}(G_{0}) be a generalized solution of problem (5), (6), where

f_{0} ∈ Ẇ_{α,2}^{0}(G_{0}),  f_{i} ∈ Ẇ_{α−2,0}^{0}(G_{0})  (i = 1, …, n),

α = −\frac{4π}{ω} − n + 4 − ε_{1}, 0 < ε_{1} < 2s − \frac{2π}{ω}.

Then u(x) can be represented in the form

u(x) = C_{1}ρ^{\frac{π}{ω}} Φ_{1}(θ) + C_{2}ρ^{\frac{2π}{ω}} Φ_{2}(θ) + u_{1},

where θ are the coordinates on the unit sphere, Φ_{1}(θ),Φ_{1}(θ) are the eigenfunctions of the Beltrami operator, and

u_{1} ∈ Ẇ_{α−2,0}^{1}(G_{0}).

Remark 1. One can readily see that all conditions of Theorems 2 and 3 are satisfied for the weights in Lemmas 1 and 2 for 0 < ε_{1} < 2s − \frac{2π}{ω}.

Bounds for |u_{1}|. Consider the cones K and K̂

K = {x | x_{3}^{2} + c ⋯ + x_{n}^{2} ≥ k^{2}(x_{1}^{2} + x_{2}^{2})},
K̂ = {x | x_{3}^{2} + ⋯ + x_{n}^{2} ≥ k̂^{2}(x_{1}^{2} + x_{2}^{2})}

and the domains

G_{1} = G_{0} \ K, Ĝ_{1} = G_{0} \ K̂.

Obviously, Ĝ_{1} ⋐ G_{1} for k̂ < k.

Lemma 3. Suppose that, in addition to the assumptions of Lemma 2, the following inequalities hold in the domain G_{1}:

|f_{0}(x)| ≤ C_{0}ρ^{\frac{2π}{ω} −2+ε},  |f_{i}(x)| ≤ C_{0}ρ^{\frac{2π}{ω} −1+ε}  (i = 1 …, n),

and let f_{0}(x) and f_{i}(x) be continuous in G_{1}, 0 < ε < s − \frac{π}{ω}.

Then

|u_{1}| ≤ C_{1}ρ^{\frac{2π}{ω} +ε},
| grad u_{1}(x)| ≤ Cρ^{\frac{2π}{ω} −1+ε}

in Ĝ_{1}.

The proof of Lemma 3 is the same as that in [4].

Remark 2. The proof of Theorem 1 is obtained on the basis of the above assertions.

Acknowledgement

The work was carried out according to Project No. 01-00 of A. M. Lyapunov French-Russian Institute.

References

1. V. A. Kondrat’ev, Boundary value problems for elliptic equations in domains with conical or angular points. (Russian) Trudy Moscow. Mat. Obshch. 16(1967), 209–292.

2. G. M. Verzhbinskiĭ and V. G. Maz’ja, Asymptotic behavior of the solutions of second order elliptic equations near the boundary. II. (Russian) Sibirsk. Mat. Zh. 13(1972), 1239– 1271.

3. V. A. Kondrat’ev, I. Kopachek, and O. A. Oleinik, Best Hölder exponents for generalized solutions of the Dirichlet problem for a second-order elliptic uquation. Mat. Sb. 131(1986), 113–125.

4. V. A. Kondrat’ev and V. A. Nikishkin, On the behavior of solutions of elliptic equations in a neighborhood of a crack with nonsmooth front. Russian J. Math. Phys. 9(2002), No. 1, 61–65.

(Received 27.03.2003)

Authors’ addresses:

V. A. Kondrat’ev Department of Mechanics and Mathematics Moscow State University, Vorob’evy gory, Moscow 119899 Russia

V. A. Nikishkin Department of Mathematics Moscow State University of Economics, Statistics and Informatics, 7, Nezhinskaya, Moscow 119501 Russia